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Parikh vectors
7RFT_1 6LDT_1 6TXW_1 Letter Amino acid
8 42 5 I Isoleucine
16 40 6 K Lycine
3 10 1 M Methionine
12 15 10 S Serine
13 25 11 V Valine
29 19 12 A Alanine
12 22 3 N Asparagine
19 28 5 D Aspartic acid
14 34 10 E Glutamic acid
17 33 8 G Glycine
4 14 4 H Histidine
3 19 4 R Arginine
12 13 7 P Proline
27 16 10 T Threonine
6 8 1 C Cysteine
5 4 0 Q Glutamine
15 37 7 L Leucine
6 15 5 F Phenylalanine
4 1 2 W Tryptophan
14 20 5 Y Tyrosine 

7RFT_1|Chains A, B|Dockerin domain-containing protein|Ruminococcus bromii L2-63 (657321)
>6LDT_1|Chain A|L-tyrosine/L-aspartate decarboxylase|Methanocaldococcus jannaschii (strain ATCC 43067 / DSM 2661 / JAL-1 / JCM 10045 / NBRC 100440) (243232)
>6TXW_1|Chains A, B|Transthyretin|Homo sapiens (9606)
Protein code \(c\) LZ-complexity \(\mathrm{LZ}(w)\) Length \(n=|w|\) \(\frac{\mathrm{LZ}(w)}{n /\log_{20} n}\) \(p_w(1)\) \(p_w(2)\) \(p_w(3)\) Sequence \(w=f(c)\)
7RFT , Knot 109 239 0.83 40 160 229 
GEETDTKIYFDASNLPAEWGTTKTVYCHLYAVAGDDLPETSWQGKAEKCKKDTATGLYYFDTAKLKSADGTNHGGLKDNADYAVIFSTIDTKSQSHQTCNVTLGKPCLGDTIYLTGGTVENTEDSSKRDFAATWKNNSDNYGPKAAITSLGHVTEGRFPIYLSRAEMVAQAIFNWAVKNPKNYTPETVADICAQVEAEPMDVYNAYAEMYATELADPAAYPDCAPLTTVATLLGVDPSG
6LDT , Knot 171 415 0.82 40 212 380 
MGSDKIHHHHHHENLYFQGMRNMQEKGVSEKEILEELKKYRSLDLKYEDGNIFGSMCSNVLPITRKIVDIFLETNLGDPGLFKGTKLLEEKAVALLGSLLNNKDAYGHIVSGGTEANLMALRCIKNIWREKRRKGLSKNEHPKIIVPITAHFSFEKGREMMDLEYIYAPIKEDYTIDEKFVKDAVEDYDVDGIIGIAGTTELGTIDNIEELSKIAKENNIYIHVDAAFGGLVIPFLDDKYKKKGVNYKFDFSLGVDSITIDPHAMGHCPIPSGGILFKDIGYKRYLDVDAPYLTETRQATILGTRVGFGGACTYAVLRYLGREGQRKIVNECMENTLYLYKKLKENNFKPVIEPILNIVAIEDEDYKEVCKKLRDRGIYVSVCNCVKALRIVVMPHIKREHIDNFIEILNSIKRD
6TXW , Knot 62 116 0.84 38 95 114 
CPLMVKVLDAVRGSPAINVAGHVFRKAADDTWEPFASGKTSESGELHGLTTEEEFVEGIYKVEIDTKSYWKALGISPFHEHAEVVFTANDSGPRRYTIAALLSPYSYSTTAVVTNP

Let \(P_w(n)\) be the set of distinct subwords (intervals) in a word \(w\). Let \(p_w(n)\) be the cardinality of \(P_w(n)\). Let \(f(c)\) be the sequence in FASTA with 4-symbol Protein Data Bank code \(c\).

\(|P_{f(7RFT_1)}(2) \setminus P_{f(6LDT_1)}(2)|=65\), \(|P_{f(6LDT_1)}(2) \setminus P_{f(7RFT_1)}(2)|=117\). Let \( Z_k(x,y)=|P_x(k)\setminus P_y(k)|+|P_y(k)\setminus P_x(k)| \) be a LZ76 style (set of subwords) Jaccard distance numerator for \(P(k)\).Hydrophobic-polar version of Sequence 1:10000001010100111011000010001011110011000101010000000101100100101001010001110001001111001000000000001011010110010101101000000000011101000000011011100110100101110100101110111011100100001001101010101011010010101010011011101001110011011110101
Pair \(Z_2\) Length of longest common subsequence
7RFT_1,6LDT_1 182 4
7RFT_1,6TXW_1 149 5
6LDT_1,6TXW_1 193 3

Newick tree

 
[
	6LDT_1:99.39,
	[
		7RFT_1:74.5,6TXW_1:74.5
	]:24.89
]

Let d be the Otu--Sayood distance d.
Let d1 be the Otu--Sayood distance d1. (This makes the 4TYN sequence AAAAAA a close match...)
A roughly speaking expected distance is \((0.85)(0.8)(\frac{654 }{\log_{20} 654}-\frac{239}{\log_{20}239})=116.\)
Status Protein1 Protein2 d d1/2
Query variables 7RFT_1 6LDT_1 145 115.5
Was not able to put for d
Was not able to put for d1

In notation analogous to [Theorem 16, Kjos-Hanssen, Niraula and Yoon (2022)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]